No crank no start with known good battery

Vehicle will rapidly click at a low volume when attempting to start the engine. Lights don't dim when trying to start it. The battery was just replaced and there are no visible signs of corrosion, although I have not checked the condition of the copper in the cables. There were no obvious signs of an imminent starter failure before this happened; it started normally until the no-start condition began. As far as I can tell, the battery terminals are screwed in tightly. Based on my research so far it seems likely that it's the starter, the solenoid, or a bad connection somewhere I've neglected to look (everywhere I checked with a multi-meter gave consistent readings) but car repair isn't my forte so I could easily be missing something. Any help is appreciated!